INNER ELASTICITY and the HIGHER-ORDER ELASTICITY of some DIAMOND and GRAPHITE ALLOTROPES Submitted by Christopher Stanley George Cousins to the University of Exeter as a thesis for the degree of Doctor of Philosophy in Physics (March 2001) This thesis is available for Library use on the understanding that it is copy- right material and that no quotation from it may be published without proper acknowledgment. I certify that all material in this thesis which is not my own work has been identi®ed and that no material is included for which a degree has previously been conferred upon me. Abstract Following a brief and selective history of elasticity, the general theory of the rÃole of relative sublattice displacements on the elasticity of single-crystalline material is elaborated in Chapter 1. This involves the de®nition of (a) rotationally-invariant inner displacements and (b) the internal strain tensors that relate those inner displacements to the external strain. The total elastic constants of such materials can then be decomposed into partial and internal parts, the former free of, and the latter involving, the inner displacement(s). Six families of inner elastic constants are needed to characterize the internal parts of the second- and third-order constants. The relation of the second- order inner elastic constants to the longwave coupling constants of lattice dynamics is shown, and a new form of secular equation for the frequencies and eigenvectors of the optic modes at the zone centre is given. In Chapter 2 the point-group symmetry implications for the inner elastic constants are explored in detail. Chapter 3 is an interlude in which the measurement of the internal strain in cubic diamond is described. In Chapters 4 and 5 the general formalism is applied to cubic and hexagonal diamond and to hexagonal and rhombohedral graphite. Space-group symmetry implications are described in detail and the formalism is extended to cover effective constants, pressure derivatives, elastic compliances and compressibilities. The allotropes are treated individuallyin terms of the Keating model in the following four Chapters. Cubic diamond is treated in Chapter 6 in terms of the original model. A shortcoming of the modelÐnon-transferability of its parameters to alternative descriptions of unit cell geometryÐis overcome by rede®ning both the Keating strain and the Keating parameters. The modi®ed Keating model is then extended rigorously and successfully to a non-cubic material, hexagonal graphite, for the ®rst time in Chapter 7. Chapter 8 presents a completely plausibleaccount of the elasticity and zone-centre optic modes in hexagonal diamond by transferring the modi®ed parameters from cubic diamond. The little that is known experimentally, the bulk modulus and three Raman frequencies, is predicted exactly. Chapter 9 extends Keating to the rhombohedral form of graphite using transferred parameters and provides a detailed picture of its transformation to cubic diamond. In Chapter 10 the relation of bond-order potentials to the Keating model is explored. An Appendix contains a generalised method of homogeneous deformation, developed to relate the computationally-friendlyin®nitesimal strain approach to the thermodynamically-rigorous ®nite strain formalism, and the associated computational protocols needed to determine all elastic and inner elastic constants, and hence all derived quantities, of the allotropes discussed. Contents Abstract 1 List of ®gures 7 List of tables 10 Foreword 11 1 Elasticity 13 1.1 A little history ..................................... 13 1.2Innerelasticity.................................... 16 1.3Macroscopicstrain.................................. 17 1.4Microscopicstrain.................................. 18 1.4.1 Occurrence and description of sublattice displacement ........... 18 1.4.2 Inner displacement .............................. 19 1.4.3 Internalstraintensors............................. 20 1.5Energyandelasticconstants............................. 21 1.5.1 Partialelasticconstants............................ 21 1.5.2 Innerelasticconstants............................ 21 1.5.3 External equilibrium of the unstressed crystal . ............... 22 1.5.4 Internal equilibrium of the stressed crystal . ............... 22 1.5.5 Composition of the total elastic constants . ............... 23 1.6 Compliances and compressibilities .......................... 25 1.7 Computational simpli®cation and sublattice tensors . ............... 25 1.8 Lattice dynamical connection . .......................... 27 1.8.1 Inner elastic constants and longwave coupling constants .......... 27 1.8.2 Thesecularequationandopticmodefrequencies.............. 28 1.9Rationale....................................... 29 2 Symmetry of inner elastic constants 32 2.1Point-groupsymmetryanalysis............................ 33 2.2 Transformation of sublattice indices ......................... 39 3 2.3 Transformation of interlattice indices ......................... 40 2.3.1 IndicesconnectingsublatticesindistinctWyckoffsets........... 41 2.3.2 IndicesconnectingsublatticeswithinasingleWyckoffset......... 43 2.4Inconclusion..................................... 44 3 Experimental interlude: the internal strain parameter of cubic diamond 46 3.1 Introduction . ..................................... 46 3.2 Inner displacement due to uniaxial stress ....................... 47 3.3 Change in structure factor due to inner displacement . ............... 48 3.4 X-ray methods .................................... 50 3.5Experimentaldetails................................. 50 3.5.1 Sample.................................... 51 3.5.2 Uniaxialpress................................. 51 3.5.3 Workingangle................................ 51 3.5.4 Beam-tailoringbytotalexternalre¯ection.................. 52 3.6Result......................................... 52 4 Inner elastic constants, internal strain tensors and zone-centre optic modes 54 4.1Symmetry....................................... 55 4.1.1 Cubic diamond and rhombohedral graphite . ............... 58 4.1.2 Hexagonal diamond and hexagonal graphite . ............... 58 4.2Internalstraintensors................................. 66 4.2.1 Cubicdiamond................................ 66 4.2.2 Rhombohedral graphite . .......................... 66 4.2.3 Hexagonal diamond . .......................... 67 4.2.4 Hexagonal graphite .............................. 67 4.3Thezone-centreopticmodes............................. 68 4.3.1 Cubic diamond and rhombohedral graphite . ............... 68 4.3.2 Hexagonal diamond . .......................... 69 4.3.3 Hexagonal graphite .............................. 71 4.4Effectiveinnerelasticconstants............................ 72 4.4.1 Cubicdiamond................................ 73 4.4.2 Rhombohedral graphite . .......................... 73 4.4.3 Hexagonal diamond . .......................... 73 4.4.4 Hexagonal graphite .............................. 74 4.5Thepressuredependenceoftheopticmodefrequencies............... 74 4.5.1 Cubicdiamond................................ 74 4.5.2 Rhombohedral graphite . .......................... 74 4 4.5.3 Hexagonal diamond . .......................... 74 4.5.4 Hexagonal graphite .............................. 75 4.6Summary....................................... 76 5 Total elastic constants, compressibilities etc. 78 5.1Anatomyofthemacroscopicconstants........................ 78 5.1.1 Cubicdiamond................................ 80 5.1.2 Rhombohedral graphite . .......................... 80 5.1.3 Hexagonal diamond . .......................... 82 5.1.4 Hexagonal graphite .............................. 83 5.2 Compliances and compressibilities .......................... 83 5.2.1 Cubicdiamond................................ 84 5.2.2 Hexagonal diamond, hexagonal graphite and rhombohedral graphite .... 84 5.3Effectiveelasticconstantsandtheirpressurederivatives............... 85 5.3.1 Cubicdiamond................................ 85 5.3.2 Hexagonal diamond and hexagonal graphite . ............... 85 5.3.3 Rhombohedral graphite . .......................... 86 5.4Summary....................................... 86 6 Cubic diamond 88 6.1 Introduction . ..................................... 88 6.2Elasticconstants,compliancesandpressurederivatives............... 89 6.3Thezone-centreopticalmodes............................ 90 6.3.1 The secular equation under stress ...................... 90 6.3.2 Phonon deformation potentials ........................ 92 6.4TheKeatingmodel.................................. 93 6.4.1 Harmonicinteractions............................ 93 6.4.2 Anharmonicinteractions........................... 97 6.5Summaryofresults..................................100 6.6Amodi®edKeatingmodel..............................102 6.6.1 Cubic diamond referred to rhombohedral axes . ...............102 6.6.2 Recasting the energy expressions ......................104 6.6.3 Modi®edKeatingparameters.........................104 6.6.4 Modi®edcubicdiamondreferredtocubicaxes...............105 6.7Summary.......................................106 7 Hexagonal graphite 108 7.1 Introduction . .....................................108 7.2 Modelling the elasticity ................................109 5 7.2.1 Appraisal of input data . ..........................109 7.2.2 Justi®cationofmodel.............................111 7.3Themodi®edKeatingmodel.............................112 7.3.1 Thestrainvariables..............................113 7.3.2 Themodelparameters............................113 7.3.3 Theenergy..................................115